Abstract

The intrinsic approach to elasticity is a variational problem that characterizes the static equilibrium state and is formulated only in terms of the unknown strain tensor field. This paper establishes an intrinsic approach to elastic cable networks undergoing large deformations. In this formulation, the potential energy function attains a global minimum at strains in any equilibrium state. It is clearly explained that both the classical displacement formulation and the intrinsic strain formulation can be derived as the Lagrange dual problems of the stress formulation, and that the difference between these two formulations stems from the variety of perturbation functions used to derive the Lagrangian in the Lagrange duality theory.

Proof

If there exists an \(i \in E\) such that \(q_{i}^{*}\) and \(\boldsymbol{v}_{i}^{*}\) satisfy \(q_{i}^{*} < \| \boldsymbol{v}_{i} ^{*} \|\), then nothing is to be proved because from (43) we see that \(L(\boldsymbol{y}, \boldsymbol{\varphi }; \boldsymbol{q}^{*},\boldsymbol{v}^{*}, \boldsymbol{r}^{*}) =-\infty \). Hence, we consider only the case in which \(q_{i}^{*}\) and \(\boldsymbol{v}_{i}^{*}\) satisfy \(q_{i}^{*} \ge \| \boldsymbol{v}_{i}^{*} \|\)\((\forall i \in E)\). Then, direct calculation yields

As a consequence of Proposition 8, the Lagrange dual problem of problem (11a)–(11c) is obtained explicitly as follows:

Maximize−∑i∈E121kiqi∗2−∑i∈Eqi∗li+∑j∈D〈rj∗,dj〉subjectto∑i∈EBij⊤vi∗=pj,∀j∈N,∑i∈EBij⊤vi∗=rj∗,∀j∈D,qi∗≥∥vi∗∥,∀i∈E.Open image in new window

Converting maximization into minimization, we obtain problem (12a)–(12d). Thus we can verify that the stress formulation in (12a)–(12d) can be obtained as the Lagrange dual problem of the displacement formulation in (11a)–(11c).

Appendix B: Second-Order Cone and Its Self-duality

The \(n\)-dimensional second-order cone (also called the Lorentz cone), denoted \(\mathcal{L}_{+}^{n}\), is defined by

We say that \(C\) is self-dual if it satisfies \(C^{*} = C\). It is known that \(\mathcal{L}_{+}^{n}\) is self-dual [1, 2, 23]. The following is an immediate consequence (see, e.g., [23, Fact 1.4.4] for a proof).